APMO inequality

If a,b,c>0 show that (a²+2)(b²+2)(c²+2)≥9(ab+bc+ca)

Prove: Let          
we just need to show that



in fact we can easily got:








Now the original inequality is equivalent to
2(xy+yz+zx)-2(xy²+yz²+zx²)-2(x²y+y²z+z²x)≤2/9 
xy²+yz²+zx²+x²y+y²z+z²x+1/9≥xy+yz+zx


by AM-GM we obtain:


                                                           

                                                                                                              Q.E.D